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Brazilian Journal of Physics

Critical Behavior of High Temperature Superconductors

C. de Calan

Published Jun 1, 2001• 0 citations
Publisher

Abstract

We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The Josephson's relation for a charged superfluid is proved without assuming the hyperscaling relation. On the other hand we discuss the dual Ginzburg-Landau model. In this dual model, due to the presence of two mass scales, a continuous family of non equivalent scalings can be defined. The relevant critical regimes are identified, and the corresponding critical exponents are predicted.

DOI
10.1590/s0103-97332001000200010

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We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The Josephson's relation for a charged superfluid is proved without assuming the hyperscaling relation. On the other hand we discuss the dual Ginzburg-Landau model. In this dual model, due to the presence of two mass scales, a continuous family of non equivalent scalings can be defined. The relevant critical regimes are identified, and the corresponding critical exponents are predicted.
We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The first part of the paper is concerned with the Ginzburg-Landau model in the zero-magnetic-field Meissner phase. We discuss the scaling behavior of the superfluid density and we give an alternative proof of Josephson's relation for a charged superfluid. This proof is obtained as a consequence of an exact renormalization group equation for the photon mass. We obtain Josephson's relation directly in the form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{t}^{\ensuremath{\nu}};$ that is, we do not need to assume that the hyperscaling relation holds. Next, we give an interpretation of a recent experiment performed in thin films of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}.$ We argue that the measured mean-field-like behavior of the penetration depth exponent ${\ensuremath{\nu}}^{\ensuremath{'}}$ is possibly associated with a nontrivial critical behavior and we predict the exponents $\ensuremath{\nu}=1$ and $\ensuremath{\alpha}=\ensuremath{-}1$ for the correlation length and specific heat, respectively. In the second part of the paper we discuss the scaling behavior in the continuum dual Ginzburg-Landau model. After reviewing lattice duality in the Ginzburg-Landau model, we discuss the continuum dual version by considering a family of scalings characterized by a parameter $\ensuremath{\zeta}$ introduced such that ${m}_{h,0}^{2}\ensuremath{\sim}{t}^{\ensuremath{\zeta}},$ where mh,0{m}_{h,0}mh,0​ is the bare mass of the magnetic induction field. We discuss the difficulties in identifying the renormalized magnetic induction mass with the photon mass. We show that the only way to have a critical regime with ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}\ensuremath{\approx}2/3$ is having $\ensuremath{\zeta}\ensuremath{\approx}4/3,$ that is, with mh,0{m}_{h,0}mh,0​ having the scaling behavior of the renormalized photon mass.
We discuss some problems related to quantum critical phenomena in superfluids and superconductors. In Ch. 1, we apply generalizations of hyperuniversality to quantum phase transitions at zero temperature. We find new universal amplitude combinations involving the superfluid density in Bose systems, as well as confirm and extend previous proposals for universal transport coefficients in two-dimensional superconducting films and magnetic-field-induced metal-insulator transitions. In Ch. 2, we apply the double-dimensionality expansion of Dorogovstev to derive renormalization-group recursion relations for a Bose fluid in a random external potential. We find a nontrivial fixed point. The onset of mean-field behavior for dimensions d > d[subscript]c = 4 is unconventional, yielding discontinuous exponents. Including positive temperatures, we give a clear picture of various crossover regimes. In Ch. 3, We give a detailed derivation of the critical thermodynamics of O(n) spin models, correct to O([epsilon] = 4 - d), using a generalization of the renormalization-group trajectory integral and noncritical matching technique introduced by Rudnick and Nelson. We especially emphasize the coexistence-curve behavior for spins with a continuous symmetry (n [...] 2), deriving detailed expressions for the renormalized spin-wave stiffness and longitudinal susceptibility.
We consider a Ginzburg-Landau model for superconductivity with a Chern-Simons term added. The flow diagram contains two charged fixed points corresponding to the tricritical and infrared stable fixed points. The topological coupling controls the fixed-point structure and eventually the region of first-order transitions disappears. We compute the critical exponents as a function of the topological coupling. We obtain that the value of the $\ensuremath{\nu}$ exponent does not vary very much from the XY\mathrm{XY}XY value, ${\ensuremath{\nu}}_{\mathrm{XY}}=0.67.$ This shows that the Chern-Simons term does not affect considerably the XY\mathrm{XY}XY scaling of superconductors. We discuss briefly the possible phenomenological applications of this model.
We discuss the critical behavior of several three-dimensional magnetic systems, such as pure and randomly dilute (anti)ferromagnets and stacked triangular antiferromagnets. We also discuss the nature of the multicritical points that arise in the presence of two distinct O(n)-symmetric order parameters and, in particular, the nature of the multicritical point in the phase diagram of high-T_c superconductors that has been predicted by the SO(5) theory. For each system, we consider the corresponding Landau-Ginzburg-Wilson field theory and review the field-theoretical results obtained from the analysis of high-order perturbative series in the frameworks of the epsilon and of the fixed-dimension d=3 expansions.
If the zero-field transition in high-temperature superconductors such as ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ is a critical point in the universality class of the three-dimensional XY\mathrm{XY}XY model, then the general theory of critical phenomena predicts the existence of a critical region in which thermodynamic functions have a characteristic scaling form. We report an attempt to calculate the universal scaling function associated with the specific heat, for which experimental data have become available in recent years. Scaling behavior is extracted from a renormalization-group analysis, and the 1/N1/N1/N expansion is adopted as a means of approximation. The estimated scaling function is qualitatively similar to that observed experimentally, and also to the lowest-Landau-level scaling function used by some authors to provide an alternative interpretation of the same data. Unfortunately, the 1/N1/N1/N expansion is not sufficiently reliable at small values of N for a quantitative fit to be feasible.
A Hartree approximation is used to study the interplay of two kinds of scaling which arise in high-temperature superconductors, namely critical-point scaling and that due to the confinement of electron pairs to their lowest Landau level in the presence of an applied magnetic field. In the neighborhood of the zero-field critical point, thermodynamic functions scale with the scaling variable [T-Tc2{\mathit{T}}_{\mathit{c}2}Tc2​(B)]/${\mathit{B}}^{1/2\ensuremath{\nu}}$, which differs from the variable [T-Tc{\mathit{T}}_{\mathit{c}}Tc​(0)]/${\mathit{B}}^{1/2\ensuremath{\nu}}$ suggested by the Gaussian approximation. Lowest-Landau-level (LLL) scaling occurs in a region of high field surrounding the upper critical-field line but not in the vicinity of the zero-field transition. For YBa2{\mathrm{YBa}}_{2}YBa2​Cu3{\mathrm{Cu}}_{3}Cu3​${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ in particular, a field of at least 10 T is needed to observe LLL scaling. These results are consistent with a range of recent experimental measurements of the magnetization, transport properties, and, especially, the specific heat of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ materials.